6.042/18.] Mathematics for Computer Science May3,2005 Srini devadas and Eric Lehman Lecture notes Expected value I The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable r defined on a sample space s is: (B)=∑R()Pr(o) To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let r be the selected student's exam score. Then Ex(r)is just the class average-the first thing everyone want to know after getting their test back In the same way, expectation is usually the first thing one wants to determine about any random variable Let's work through an example. Let r be the number that comes up on a fair, six-sided ie. Then the expected value of R is Ex 1+2·+3:+4+6. 1 6 variable might never actually take on that value. You cant roll a 3) on an ordinary die/? This calculation shows that the name"expected value"is a little misleading the rando 1 Betting on Coins Dan, Eric, and Nick decide to play a fun game. Each player puts $2 on the table secretly writes down either heads "or tails". Then one of them tosses a fair coin $6 on the table is divided evenly among the players who correctly predicted the outcome of the coin toss. If everyone guessed incorrectly, then everyone takes their money back. After many repetitions of this game, Dan has lost a lot of money- more than can b explained by bad luck. What's going on? A tree diagram for this problem is worked out below, under the assumptions that everyone guesses correctly with probability 1 /2 and everyone is correct independently� � � 6.042/18.062J Mathematics for Computer Science May 3, 2005 Srini Devadas and Eric Lehman Lecture Notes Expected Value I The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable R defined on a sample space S is: Ex (R) = R(w) Pr (w) w∈S To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let R be the selected student’s exam score. Then Ex (R) is just the class average— the first thing everyone want to know after getting their test back! In the same way, expectation is usually the first thing one wants to determine about any random variable. Let’s work through an example. Let R be the number that comes up on a fair, six­sided die. Then the expected value of R is: � 6 1 Ex (R) = k 6 k=1 1 1 1 1 1 1 = 1 · + 2 · + 3 · + 4 · + 5 · + 6 · 6 6 6 6 6 6 7 = 2 This calculation shows that the name “expected value” is a little misleading; the random variable might never actually take on that value. You can’t roll a 3 1 2 on an ordinary die! 1 Betting on Coins Dan, Eric, and Nick decide to play a fun game. Each player puts $2 on the table and secretly writes down either “heads” or “tails”. Then one of them tosses a fair coin. The $6 on the table is divided evenly among the players who correctly predicted the outcome of the coin toss. If everyone guessed incorrectly, then everyone takes their money back. After many repetitions of this game, Dan has lost a lot of money— more than can be explained by bad luck. What’s going on? A tree diagram for this problem is worked out below, under the assumptions that everyone guesses correctly with probability 1/2 and everyone is correct independently